Dynamics. 

 or, since 



Trig. 27. cos a sin (a e) = (sin (a -f- a e) sin (a a -j- c) ). 



4 /& i (sin (2 e) sin e) = c cos e, 

 and 



2 A 2 /i sine 



sin (2 a e) = - - -f c, 



cos e cos e 



which may be given by 'the following construction. 



Having raised upon AM the indefinite perpendicular AE ; 



from the middle D of ./^ = 4 /t, we erect upon w^^f the perpen- 

 dicular Z), cutting AE in some point E, from which as a centre, 

 and with a radius equal to EA, we describe the arc JlNN'K ; 

 having produced PM till it meets this arc in the points JV, JV', if 

 we draw the lines ANZ, AN'Z', these will be the directions iu 

 which the projectile being thrown with a velocity due to the 

 height h, it will in either case fall upon the point M. 



Indeed it will be readily seen that the angle EAD of the 

 right-angled triangle ADE is equal to MAP. Therefore, since 



AD = 2 /t, ED = ^-r4 5 an d, since AP = c, we shall have, 



cos 



cos e 

 consequently, 



^ sin(2a e) = i:/. 

 cos e 



But in the same triangle ADE, AE = ^ ; therefore, 



COS 6 



AE sin (2 a e) = /. 



Let the arc KNA be produced till it meets, in Cr, the vertical GE, 

 and from the points JV, JV 7 , draw the perpendiculars JVL, JNPX 7 . 

 In the triangle JVEL, we have 



WE : JVL, or AE : El : : 1 : sin NEG ; 

 whence, 



AE sin JVG = El; 



accordingly, 



